Bolzano Theorem (BT)

Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such that f(a) and f(b) are of opposite signs. Then there exists a number x0[a, b] with f(x0)=0.

Intermediate Value Theorem (IVT)

Let, for two real a and b, a < b, a function f be continuous on a closed interval [a, b] such that f(a)<f(b). Then for every y0 such that f(a)<y0<f(b) there exists a number x0[a, b] with f(x0)=y0.

Clearly BT is only a special case of IVT. However it's interesting that the more general IVT could be deduced from its special case, BT. Indeed, let f(a)<y0<f(b). Introduce a new function g(x)=f(x)-y0. Then it follows that g(a)=f(a)-y0<0 whereas g(b)=f(b)-y0>0. Therefore, g(a) and g(b) are of opposite signs. Additionally, g is continuous wherever f is. In particular, g is continuous on [a, b] and thus satisfies the conditions of BT. Therefore, there exists x0[a, b] such that g(x0)=0. Written explicitly this says f(x0)-y0=0 or, finally,

f(x0)=y0

Q.E.D.

A complete proof of the Bolzano Theorem appeares elsewhere.


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